Nonuniformly optimally spaced antenna array

ABSTRACT

The object of this invention is to provide a new nonuniformly spaced antenna array, where the most optimum positions of the array elements, and their corresponding amplitudes, are systematically determined by a rigorous synthesis technique of the given radiation pattern, and/or its requirements and specifications, in order to achieve it with the minimum possible number of array elements. This new array will be designated the Nonuniformly, Optimally Spaced Antenna Array, or in short, the NOSA Array.

1451 Dec. 18, 1973 NONUNIFORMLY OPTIMALLY SPACED ANTENNA ARRAY [76]Inventor: Hillel Unz, c/o Electrical 5 Engineering Dept. University ofKansas, Lawrence, Kans. 66044 221 .Filed: Jan. 17, 1972 211 Appl.No.:218,495

3,553,706 l/l97l Charlton 343/844 3,262,! l5 7/1966 Macalpine 3,182,3305/1965 Blume 343/844 Primary ExaminerEli Lieberman Attorney- Hillel Unz5 7 ABSTRACT The object of this invention is to provide a newnonuniformly spaced antenna array, where the most optimum positions ofthe array elements, and their corre- [52] US. Cl. 343/844, 343/719sponding amplitudes, are systematically determined by [51] Int. Cl.l-l0lq 21/00 a rigorous synthesis technique of the given radiation [58]Field of Search 343/719, 8 4, 85 pattern, and/or its requirements andspecifications, in 3/854 order to achieve it with the minimum possiblenumber of array elements. This new array will be designated [56]References Cited the Nonuniformly, Optimally Spaced Antenna Array,

UNITED STATES PATENTS or in short, the NOSA y- 3,l30,4l0 4/1964 Cutleber343/844 6 Claims, 2 Drawing Figures n 0 l 2 3 4 5 b 7 8 TABLE ANONUNIFORMLY OPTIMALLY SPACED ANTENNA ARRAY The general idea of thenon-uniformly spaced antenna arrays was invented by Unz in 1955, and waspublicly proposed and published for the first time in-his University ofCalifornia (Berkeley, I956) doctoral dissertation, where the firstsignificant work on the subject was reported. From 1960 on manyadditional contributions have been made on non-uniformly spaced arraysby numerous authors, and references to most of them may be found inrecent books on antennas. However, the main synthesis problem of thenon-uniformly spaced arrays, namely, finding the most optimum positionsof the elements of the array in order to produce a given specifiedradiation pattern, has not been solved rigorously to-date. This isprimarily due to the great difficulties in the solution of this highlynon-linear problem. The trial and error computer techniques and theother pseudo-optimum synthesis methods suggested so far are of verylimited utility, and are almost impossible to employ with arrays of alarge number of elements. Thus, the non-uniformly spaced arrays cannotbe technically designed at the present time to their full potentialadvantage, if at all, and therefore are not generally used.

The object of this invention is to provide a nonuniformly spaced antennaarray, where the array elements are distributed in the most optimumpositions along the axis of the array, with reference to the givenradiation pattern requirements and specifications. The optimaldistribution of the elements along the array axis is determined so thatthe Nonuniformly Optimally Spaced Antenna Array (the NOSA Array) of thisinvention will require a smaller number of elements, than an equivalentuniformly spaced array, giving the same required approximate radiationpattern, or its specified required characteristics, such as the sidelobelevel, and/or beamwidth, and/or gain, etc. Alternatively, the NOSA Arraywill give a better performance in its radiation pattern and itscharacteristics than an equivalent uniformly spaced array of an equalnumber of elements. A rigorous synthesis technique is provided for thesystematic determination of the most optimum positions of the el' ementsin the Nonuniformly Optimally Spaced Antenna Array (the NOSA Array), andtheir corresponding amplitudes, in order to accomplish the radiationpattern given specifications with the minimum possible number ofelements in the array. Thus this invention will provide a substantialsavings in the number of elements required in an antenna array for agiven performance. While the rigorous synthesis technique describedlater is for the symmetric linear NOSA Array, the invention by no meansis limited to this particular case. It could be extended quite easily tothe non-symmetric NOSA Array. Furthermore, since the NOSA Array is thebuilding stone for any number of other more sophisticated arrays, thisinvention covers all other arrays, where they include in whole or inpart the NOSA Array. Thus this invention covers, but is not limited tothe following arrays, if they include in whole or in part the NOSAArray: symmetric and non-symmetric arrays, linear, two and threedimensional arrays, arrays on curved surfaces, transmitting, receivingand retrodirective arrays, arrays of arrays, correlation arrays,steerable beam, scanning arrays and phased arrays, active and adaptivearrays, time modulated arrays, radioastronomy arrays, broadband arraysand frequency independent arrays, superdirective arrays, slot arrays,omnidirectional arrays, broadside arrays, end-fire arrays, resonant andnonresonant arrays and many others.

The invention is illustrated by the accompanying drawing in which:

Table A gives the seventeen (l7) complex Fourier series coefficients f,=f specified from the given real and symmetric radiation pattern to beapproximated by the symmetric linear antenna array. FIG. 1 and Table Agive the seventeen l7) element half-wavelength uniformly spaced antennaarray which will give the specified Fourier coefficients f listed inTable A. The synthesis has been done in accordance with well establishedprocedures for uniformly spaced arrays. FIG. 2 and Table A give the nine(9) element Nonuniformly Optimally Spaced Antenna Array (the NOSA Array)which will give the same specified Fourier coefi'lcients f listed inTable A. The systematic determination in FIG. 2 of the most optimumarray element distribution and their corresponding amplitudes for thegiven Fourier coefficients in Table A, have been accomplished by usingthe rigorous synthesis technique described below. Thus the NOSA Array inFIG. 2 will give the same required performance as the half-wavelengthuniformly spaced antenna array in FIG. 1, but with a substantial savingof almost half of the number of the elements required by the standarduniformly spaced array. A substantial saving of the number of theelements of the array for equivalent performance is of essentialimportance especially in arrays with a large number of elements.

The rigorous synthesis technique for the symmetric linear Non-uniformlyOptimally Spaced Antenna Array (NOSA Array) for a given real andsymmetric radiation pattern or its complex Fourier series coefficients ff, real specified in Table A, in order to determine the most optimumpositions of the elements of the NOSA Array, and their correspondingamplitudes, in FIG. 2, is described as follows:

The radiation pattern F (0) of a linear array with nonuniform spacingsof the elements is given by:

F) 2 A ikd m a where u 1r sin 0 11' 5 15 11') and x,=d,/M2 gives thedistance of the array element p from the origin in terms of halfwavelengths.

For the synthesis problem the given radiation pattern F (u) may beexpanded in a complex Fourier series, for n integer:

For the particular case of half wavelength d p M2 uniformly spacedlinear array one has x, p integer in (Lb), which then has the same formas Q a). Thus, each element of the uniformly spaced array corresponds toone complex Fourier series term A =f,,, and F(u) in general may beapproximated to any degree of accuracy by a finite number of arrayelements, corresponding to the same number of complex Fourier seriesterms.

in case of a non-uniformly spaced array, x p) in (1b) has to bedetermined, as well as A,,, for the best approximation of F(u).Substituting (Lb) in (2.1;) one obtains after integrating:

Using the trigonometric indentity and taking sin mr O and cos mr (1 Y,one has:

sin(n x,,)rr sin mr cos x 'n' cos m-r sin x n' (i )"''sin x n' and oneobtains from (3.a):

Taking the array element p of the amplitude A to be at the origin x 0,one obtains from (4.a) and (4.b):

Let us assume that the given radiation pattern F(u) is a real andsymmetric function F(-u) F(u). Thus one finds from (211):

Let us further assume that the non-uniformly spaced array has an oddnumber of elements and is symmetric with respect to the element A at theorigin x,, 0. Using (4.b) one obtains:

Substituting (6.b) in 7.11 and (7.b) one obtains after rearranging:

From (8.11) one finds thatf =f,, as in (6.a). Taking in (8.a) D, B,,x,,,N n, y X92 and using (4.b) one obtains:

We will discuss here only the solution of (9.a) for the symmetricnon-uniformly spaced array.

In equation (9.0) we have two sets of unknowns, y and D,,, which have tobe solved, and thus give us x, and A, from (9.1)). Writing (9.0)explicitly for s terms and multiplying by the common denominator on bothsides, one obtains for N n 2 l:

where (11) for given f, is a linear equation for D, and a non-linearequation for y,,. Opening the brackets on both sides and rearranging indescending powers of N, one obtains:

---+ s (yr y2 y3 -y.. 1) (m) Since y, and D are the unknowns to befound, and thus 04,, and B defined above are also the unknowns to befound. Thus (l2.a) couldbe rearranged and rewritten as follows for)! n 21:

or alternatively, dividing (13.12) by N one obtains:

alternatively (l3.b) may be rewritten in the following form:

Equation (1.3.a) or (l3.b) or (13.0) represents a linear equation with2s unknowns, namely, s unknown terms of a,,, and s unknown terms of [3One may rewrite one of the equations (13), i.e. (13.0), 2s times for thegiven 2s Fourier coefficients f,, of the given radiation pattern F(u),namely for (f f f, .f where N= n 1. Thus one obtains 2; linear equationswith 2s unknowns 01,, and B The method of solving Zr linear equationswith 2s unknowns by the use of determinants and by well established andmodified computer techniques is ye ry well known Thus if (f f ,f aregiven, one may solve l 3.c) for a (a,, a 11,), by determinants with theaid of a computer. As it will be shown later, there is no need to solvefor [3,.

Once (a a 01,) are thus found, one should solve the s non-linearequations (l2.b) for (y,, y .,y It is well known that a polynomialequation with the solutions y y,, y y y y, could be written in the form:(yyi)(yyz)(yya)----(y-y8)= (14 11) From the left hand side of (l l) and(12.0) and from (l2.b) one finds by inspection and by multiplying andopening the brackets of (l4.a) one obtains: ya,y"+a y a y +(l)a,=0

But the coefficients ((1,, a 01,) have been solved above. Thus thepositions of the elements x, along the axis of the non-uniformly spacedarray could be found as the roots of the polynomial equation l4.b) y,=x,, Finding the roots of the polynomial equation (l4.b) is a wellestablished process with the aid of a computer. The positions x thusfound from the roots of the algebraic equation (l4.b) represent theoptimum positions of the symmetric NOSA Array elements.

Once the s roots y have been found from (l4.b), one may substitute themin (9.a) and rewrite it s times for (f,,f ,fl). Thus one obtains slinear equations with s unknown D,,, which could be solved by usingdeterminants and well known computer techniques. With all y, and D thusfound, one may use (9.b) to obtain: x,, A, D,,/(x,, sin x qr) (15 1} andthus the position of each element x, and its amplitude A, are found forall the side elements of the symmetric non-uniformly spaced array. Oncethis is known, one may use (8.b) and (9.b) to find the amplitude of thecenter element A at x O:

and the synthesis problem for a given F (u) is solved for thenon-uniformly optimally spaced array.

The details of the procedure of the rigorous synthesis technique for thedetermination of the most optimum positions of the elements of the NOSAArray and their corresponding amplitudes for the symmetric case in FIG.2 are given as follows:

Assuming that 2s Fourier series coefficients of f, have to be used, onerewrites the linear equation (13.0) of 2s unknowns 01,, and B, for atotal of 2s times, using the given f, for l 3 n s 2s and taking N n The2s linear equations with 2s unknowns are solved for all a (a a 0,) bythe use of well known computer techniques.

Using the 01,, coefi'icients found previously, one obtains the algebraicequation l4.b) of order s. The positive real roots of the algebraicequation (l4.b) are found by using well known computer techniques, and

they represent the optimum positions of the symmetric NOSA Arrayelements, where x, V'yj.

Substituting the value of y, found above in (9.0), and rewriting it 5times for different f, for l n 5 s, taking N n one obtains s linearequations for s unknowns D,,, which can be solved by the use of knowncomputer techniques. Once D and y, have been solved for the symmetriclinear NOSA Array, obe obtains the position of the p element x,,, andits amplitude .4 by using (91;). The amplitude A,, of the center elementat x, O is found by using (l5.b). Using the above rigorous synthesistechnique, the symmetric linear NOSA Array in FIG. 2 was technicallydesigned from the given complex Fourier co-efficients in Table A. Thenine (9) element NOSA Array in FIG. 2 is equivalent with its requiredperformance to the seventeen 17) element half wavelength uniformlyspaced array given in FIG. 1. Both the uniformly spaced array in FIG. 1and the nonuniformly optimally spaced array in FIG. 2 give almost thesame identical radiation pattern which has been prescribed and definedby the f, complex Fourier series coefficients given in Table A.

Let us assume that in prder to approximate a given symmetric and realfunction radiation pattern F (u) to a certain given degree of averagemean square approximation, one needs to have a symmetric linear arraywhich will give exactly the values of the complex Fourier coefficientsf,, for s n s 2M. If we chose a symmetric uniformly spaced array of halfwavelength spacings, we determine in advance the element positions andwill need, in accordance with equation a total of(4M l antenna elements,for the symmetric array with the center element. However, if we choose anonuniformly spaced symmetric linear array, where both the positions ofthe elements and their corresponding amplitudes are to be determined forthe best approximation, we have twice as many degrees of freedom, and wecould approximate 2M Fourier coefficients by using only M radiationelements in accordance with (l3). Thus the nonuniformly spaced symmetriclinear array with the center element would require a total of only(2M l)non-uniformly spaced antenna elements in order to achieve approximatelythe same radiation pattern; therefore one has for a symmetric lineararray, where f for 0 s n s 2M is given:

No. of elements uniform 4M+ 1 1 N 2 No. of elements nonuniform 2M 1 2M 1One needs therefore half the number of elements in the Non-uniformlyOptimally Spaced Antenna Array (the NOSA Array) as compared to auniformly spaced array giving the same approximate radiation patternwith all its side lobes. In this statement we assume that all the rootsin the polynomial equation (l4.b) are positive and real: thus the resultdepends on all the coefficients a,,, and on the given FourierCoefficientsf through the linear equation (l3.c). The results of such anumerical example for a given prescribed set of Fourier coefficients f,,are given above.

Furthermore, one should check an additional number of Fourier arraycoefficients f,, for n 25, which have not been approximated, but whichalso may be found from a given radiation pattern F(u), as well as thefinal resulting radiation pattern F(u) given by the NOSA Array and itselements A, at positions x, in accordance with equation (lb). In casethe approximation requirements of the NOSA Array radiation pattern withrespect to the given prescribed radiation pattern are not acceptable,one may repeat the process by adding one more antenna element on eachside of the NOSA Array and repeat the process again, and check again forthe approximation acceptability. The number of the elements in the NOSAArray will be still smaller than an equivalent uniform array, but byless than the factor of two given in (16). The final result depends oneach indivudal or class of given radiation patterns.

. half wavelength uniformly spaced linear array giving the sameprescribed radiation pattern with all its sidelobes.

Since the non-uniformly optimally spaced array described in theforegoing specifications could be used as the basis for building moresophisticated arrays with specific or minimax performance indicesrequirements, this invention covers, but is not limited to, arrays withspecific requirements on the radiation patterns, impedance, bandwidth,mutual coupling, gain and directivity, polarization, noise temperature,signal to noise or interference ratio and any other specific indices, ifthe array includes in whole or in part certain definitive aspects ofthis invention as a part of its analysis or synethsis. This inventionalso covers this array, when the elements used in the array are ofdifferent types, including but not limited to, dipoles, slots, horns,apertures, parabolic reflectors, dishes and many others. Many of thedesign techniques in prior art developed for electromagnetic arrays canand has been applied with modification to acoustic arrays, seismicarrays and arrays in other fields. Thus this invention of non-uniformlyoptimally spaced array described in the foregoing specifications covers,but is not limited to, all such arrays or combination of them in otherfields for any purpose. This invention covers non-uniformly optimallyspaced acoustic arrays and sonar arrays, used with acoustic elements ofany type, when used under water or above water for whatever purpose.This invention covers non-uniformly optimally spaced seismic arrays,used with seismic elements, geophones, seismometers seismographs, orother elements, when used underground for whatever purpose. Thus theforegoing invention of non-uniformly optimally spaced array covers anyarray of any shape and size, and used with any kind of elements for anypurpose, provided that the invention described here will be used therein whole or in part, or will be used there as a part of its analysis orsynthesis during the design procedure.

While in the foregoing specification, I have set forth certain detailsof the Nonuniformly Optimally Spaced Antenna Array (the NOSA Array) andits rigorous synthesis technique for a specified radiation pattern, forthe purpose of illustrating one mode of the invention, it will beunderstood that such details may be varied widely by those skilled inthe art without departing from the spirit of my invention, and it istherefore aimed to cover all such changes and modifications in all areasof endeavor where arrays are used, as fall within the true spirit andscope of this invention.

What I claim is l. A method for optimizing the spacing and ampli- D forn0 where f, is the n-th complex Fourier series coefficient of the givenradiation pattern, N=n y x, where x,, is the distance of the p-thelement of the array from the origin in terms of half wavelengths, andD, A r: sin x,,1r where A is the amplitude of the p-th element of thearray, the said method comprising the selecting of the optimum positionsof the individual elements from the s roots y=y,,=x,, of the followingalgebraic equation: 3 -04 y y -a y" .+(l) a, =0 where the coefficients01,, are determined by the solution of the following 2s linear equationsfor the given f coefficientsz I if i S 1) aq. 1 S om n once the optimumpositions x are determined, the optimum amplitude of each element isfound from (A) by solving 5 linear equations for the unknowns D,, theamplitude of the center element A, at the origin x 0 being given by:

2. A method for optimizing the spacing and the amplitude of theindividual elements in a non-uniformly spaced non-symmetric array, ascompared to a conventional uniformly spaced array, in order toaccomplish a given radiation pattern with the minimum number ofelements, said method accomplished essentially as in claim 1, except itis based on the following relationship for a non-symmetric array:

where B, A, sin x,,1r and (E) is almost identical in form to (A) inclaim 1.

3. An arrangement of a set of elements in a nonuniformly optimallyspaced array, as specified in claim 1, in which each element is anelectromagnetic antenna radiator or receiver, like dipole, slot, horn,aperture, dish, parabolic reflector, etc. or a combination of them.

4. An arrangement of a set of elements in a nonuniformly optimallyspaced array, as specified in claim 1, in which each element is anacoustic or sonar radiator or receiver, or a combination of them, usedunder water or above water 5. An arrangement of a set of elements in anonuniformly optimally spaced array, as specified in claim 1, in whicheach element is a seismic radiator or receiver, like geophone,seismometer, seismograph, etc. or a combination of them, used underground or above ground.

6. An array comprising a grouping of subarrays each designed inaccordance with the method set out in claim 1.

1. A method for optimizing the spacing and amplitude of the individualelements in a non-uniformly spaced array, as compared to a conventionaluniformly spaced array, in order to accomplish a given radiationpattern, and/or its requirements and specifications, with the minimumnumber of elements, said method based on the following relationship fora symmetric linear array:
 2. A method for optimizing the spacing and theamplitude of the individual elements in a non-uniformly spacednon-symmetric array, as compared to a conventional uniformly spacedarray, in order to accomplish a given radiation pattern with the minimumnumber of elements, said method accomplished essentially as in claim 1,except it is based on the following relationship for a non-symmetricarray:
 3. An arrangement of a set of elements in a non-uniformlyoptimally spaced array, as specified in claim 1, in which each elementis an electromagnetic antenna radiator or receiver, like dipole, slot,horn, aperture, dish, parabolic reflector, etc. or a combination ofthem.
 4. An arrangement of a set of elements in a non-uniformlyoptimally spaced array, as specified in claim 1, in which each elementis an acoustic or sonar radiator or receiver, or a combination of them,used under water or above water.
 5. An arrangement of a set of elementsin a non-uniformly optimally spaced array, as specified in claim 1, inwhich each element is a seismic radiator or receiver, like geophone,seismometer, seismograph, etc. or a combination of them, used underground or above ground.
 6. An array comprising a grouping of subarrayseach designed in accordance with the method set out in claim 1.